Assume a function called $f(x)$ , Then all of us know that of we draw $f(x+a)$ it will be a transformation to the left or right and $f(x)+b$ to up or down.
But when I drew floor function on Desmos online graphing I found something a little bit different.
https://i.stack.imgur.com/gPFER.jpg
As you see that the black function is exactly on the purple one and that confusing me , Here $f(x+a)=f(x)+a$ , And we are just rising the function up .
To be more specific I need you to explain these question :
$1.$ Why when we add a number in the floor function notation $f(x+a)$ or $[\frac {x}{2}+1]$ it rise the function.
$2.$How can transform the function to the left just a unit.
The fact that $\lfloor f(x) + 1 \rfloor = \lfloor f(x) \rfloor + 1$ for any function $f$ should be fairly obvious, if you think about what the floor function does: round down to nearest integer. (You can replace "$+1$" here with "$+n$" for any integer $n$, but it doesn't work for non-integers.)
So for your particular example $f(x) = \frac{x}{2}$, you get $\lfloor \frac{x}{2} + 1 \rfloor = \lfloor \frac{x}{2} \rfloor + 1$, and the graph is moved one unit upwards.
But that's also equal to $\lfloor \frac{x}{2} + 1 \rfloor = \lfloor \frac{x+2}{2} \rfloor = \lfloor f(x+2) \rfloor$, so it's the same thing as moving the graph two units to the left.
If you want to move the graph one unit to the left, take $\lfloor f(x+1) \rfloor = \lfloor \frac{x+1}{2} \rfloor = \lfloor \frac{x}{2} + \frac{1}{2} \rfloor$ instead.